1. Field of the Invention
The present invention relates generally to a method of generating a finite element mesh (FEM) for finite element analysis (FEA), and in particular, to a method and apparatus for generating a three-dimensional FEM.
2. Description of the Related Art
A finite element method is one of the techniques for numerical analysis of approximation of differential equations. The finite element method is generally used in fields which deal mainly with the structural dynamic analysis of objects, such as mechanical engineering. The finite element method finds wide use in applications for analysis of the strength and transformation of structures, fluid flow analysis, and electromagnetic field analysis. Particularly, it is preferable to analyze the structures of parts in a design process in order to save design cost and marketing time in manufacture of various electric, electronic products and their component parts.
To meet design engineers' demands, CAD (Computer Aided Drafting) is used as a basic FEA tool. In addition, Pro-Engineer, CATIA, and IDEAS are utilized as three-dimensional modeling tools.
The finite element method decomposes a problem domain to be analyzed regarding a structure, machine or part of interest into elements of a predetermined volume as a pre-process of FEA. This is called meshing. Particularly, for structural analysis of a three-dimensional model, an FEM is generated by partitioning the area that the three-dimensional model occupies into a plurality of fine elements called solid elements.
A solid element is a simple geometrical form, for example, in a tetrahedral or hexahedral shape. Solid elements may be divided into a tetrahedral element, a hexahedral element, a prism element, and a pyramid element according to their configurations. Despite its ability of freely representing any three-dimensional shape, the tetrahedral solid element is of low quality in analysis accuracy and requires complex computation because too many solid elements are produced. Thus, a hexahedral mesh is usually generated using hexahedral solid elements (or prism solid elements) tto increase analysis accuracy or computation efficiency.
FIGS. 1A to 1E illustrate sequential FEM models generated in a conventional FEM generating method. A cup is FEM-modeled three-dimensionally, by way of example. An initial cup-shaped model can be a CAD model. Traditionally, to generate a three-dimensional FEM of the cup geometry, an operator generates a solid mesh for the bottom of the cup by solid meshing, as illustrated in FIG. 1A. Referring to FIGS. 1B and 1C, the wall of the cup is solid-meshed by stacking elements of an appropriate shape on the bottom solid mesh. Referring to FIGS. 1D and 1E, a solid mesh is generated for the handle of the cup by applying elements of an appropriate shape to the wall solid mesh. Thus, a final cup solid mesh is completed.
As described above, the operator generates solid elements by manually defining nodes that form each solid element one by one and generates a final three-dimensional solid mesh in the conventional FEM generating method. This is a very complex task. Therefore, a long time is required and an experienced operator is needed to generate good-quality elements. For a car engine, for example, a modeling time taken to generate a hexahedral FEM from a CAD model can take anywhere from 3 to 6 months. In the case of a general electronic appliance, a model time between one day and thirty days is typically required.
To reduce the time required for generation of a three-dimensional FEM, various techniques have been developed to automate the FEM generation with the aid of a computer. Examples of the techniques are U.S. Pat. No. 5,768,156 entitled “Connectivity-Based, All Hexahedral Mesh Generation Method and Apparatus”, and U.S. Pat. No. 6,578,189 entitled “Hexahedral Mesh Generation Method and Device”. The above FEM generation methods have limitations in generating a good-quality FEM because they require a large volume of computation and many elements of relatively small sizes.